simple algebra to the 3d mandelbrot

[kujonai]

the substraction stops when a pile of the 3 is zero, because of in the plane are necesary only 2 coordenates, the eisenstein numbers are alternative form of the 3-signed arithmetic, more exactly a kind of polinomial form of 3-signed arithmetic( at least i think that), and is too a particular case (n=3) , the only difference is that the mod-operator is tranfered to the exponent. like it follow:

 0]1+1]1+2]1=0                1+v+v^2=0                   to  remember:  0]1=+1     and v is a complex cube root of unity

(1]1)^1=1]1                     v^1=v

(1]1)^2=2]1=-1]1-1         v^2=-v-1

(1]1)^3=3]1=1                v^3=v^0=1

http://en.wikipedia.org/wiki/Eisenstein_integer

and simply, the 4-level of musean hypernumber w........  w=-v

http://en.wikipedia.org/wiki/Musean_hypernumber

regards

[jehovajah]

 0]1+1]1+2]1=0                1

I have seen this rule on the polysigned site http://www.bandtechnology.com/PolySigned/PolySigned.html

But think it is one example of many ways to proceed with this unary operator exploration. It has its origins in geometrical constraints. In general unary operetors are defined for operating on one thing at a time. We must define binary trinary, ... many argument operators to deal with each signals combination with the others. The law of identity so called is such a many argument operator . I am not assuming that unary operators only act on the plane geometrically,and i am not assuming that the axes which may be used to illustrate a signal collection is the only way to arrange a signal collection. The chap at polysign points out the need to be clear what one is doing. So addition and subtraction may make sense within a signal group but across the signals he uses the notion of superposition.

The law of identity is thus placing equal magnitude signals in superposition we can define as eliminating the signals of that magnitude. What superposition is in a concrete example has to be explored for each mod(n).

With regard to complex nos or the set mod(4) should define the unary operator i in the plane (rotating by ) each time, but we still have to define a ninary operator for the superposition of signals and the structure of transformation of (a, b)

[TimGolden]

As I see it Kujonai may still be developing his format.
I would pose the question:
   Are you attempting to generalize sign?
If so then I believe it is important to first accept that the real number is two-signed and that its elemental format is
   s x
where s is one of two signs; either - or +, and x is a magnitude. Thus a three-signed number can reuse this format with the addition of a new sign. I have chosen to remain consistent with the real number, though in higher sign systems the meaning of '+' then changes. Still, the mnemonic meaning of the signs is consistent, with the number of strokes it takes to draw each sign consistent with its modulo mechanics under product:
   -, +, *, #, ...
with one stroke, then two strokes, then three strokes, then four strokes, and then I run out of symbols, but this is acceptable, since upon getting that far it becomes apparent that a general component form can take over:
   ( 1.1, 1.2, 1.45, 2.34, 0.23 )
being a five-signed number in its most concrete instance. The position of the components are the sign positions. Heading back down to the real number we see that:
   ( 1, 1 ) = 0
which is simply stating that
   - 1 + 1 = 0
and it is this property; the balance of the signs; which yields dimension since the next form (P3, the three-signed numbers) takes the behavior
   ( 1, 1, 1 ) = 0
such that a simple expression in P3 as
   - 1.2 + 2.3
does not any longer cancel as did a real valued( two-signed; P2) expression. Hence the three-sign numbers are two-dimensional.
The four-signed numbers are three dimensional. P3 are the complex numbers, entirely consistent with the definition of the reals, merely up one in sign. Someone who comprehends this should be able to appreciate polysign, even just in the context of definition of real and complex numbers. They are united through polysign. The simplex coordinate system is natural to polysign. There is no need of any cartesian product construction. The generalization of sign immediatley yields dimensional behavior.

I see Kujonai somewhat taking a more complicated approach which may have some validity as the embedding of say a two-sign system within a three-sign system is as appropriate as embedding a line in a plane. Still, I would argue that the fundamental layout that I've presented lays beneath such an approach and will aid in the construction of that higher level concept.

 - Tim
 

[Tglad]

Hi Tim. I would like to say that I really like your polysign construction, and like the idea of basing the coordinates on the simplex, of which there's one in every dimension.
You may also have noticed that P3 lacks some of the oddities of complex numbers. The +i and -i are completely interchangeable in complex numbers so the + means something very different to + on the reals. Also, unlike the reals, the complex units 1,-1,+i,-i are not equidistant, whereas the polysign units are always so.

However, I think the problem is the product function. I can't comment for >4 signs but the 4-signed numbers are basically just being complex numbers on a plane. That is why they produce the extruded mandelbrot here: http://www.fractalforums.com/theory/polysigned-mandelbulbs/?action=post;quote=9059;num_replies=4;sesc=cf68cf6d4b24e76361b129afd4de4d84

I think a probable better product rule would fold the 3d space onto itself, as show in this video http://www.fractalforums.com/3d-fractal-generation/a-new-3d-mandelbrot-like-fractal/?action=post;quote=9787;num_replies=10;sesc=cf68cf6d4b24e76361b129afd4de4d84
I'm not sure exactly the math but, the basic cases would be roughly like this:
(+)(+) = +
(-)(+) = -
(*)(+) = *
(#)(+) = #

(-)(-) = +
(*)(*) = +
(#)(#) = +

(-)(*) = #
(*)(#) = -
(#)(+) = *

[kujonai]

the 4-signed and complex use mod 4, i tinhk the differences are in the substraction rules,
in the equation x^n-1

the complex see the answer in the regular poligons
the poly-signes see the answer in the  regular simplex

regards   

[TimGolden]

In the mod-2 system we see that '+' as sign=2
makes sense even within the product.

   - : sign = 1
   + : sign = 2

To me the ultimate solution is the zero sign:
   @ : sign = 0
This then becomes the universal superposition sign, which will not
change with a change in level succh as going from mod2 to mod3.
Unfortunately the '+' symbol will not maintain its mnemonic meaning in
generalized sign. The '@' sign takes its place, the '@' so chosen
because it is like a zero but cannot be confused with zero. With a zero
sign the sign product definition can be defined as modulo superposition
without the complication of the plus sign as sign=2. To me the sign
symbology should be clean mnemonically. This means
   @ : sign = 0 (like zero)
   - : sign = 1 (one stroke)
   + : sign = 2 (two strokes)
   * : sign = 3 (three strokes)
   # : sign = 4 (four strokes)
Anyhow we run out of decent symbols. There is little value in quibbling
over formats. Many formats are appropriate; from
   a + b i + c j
type of nonorthogonal unit vector to
   ( a, b, c )
type of coordinate format to
   @ a - b + c
polysign format to
   + a $ b # c
in opuesto aritmeticas format.

We have to coexist. In order to do so I suggest that in our
communication we each abandon our signs as much as possible and settle
into the coordinate representation where in the mod-4 system we see the
product
   (1,0,0,0) (z11,z12,z13,z14) = (z11,z12,z13,z14)
   (0,1,0,0) (z11,z12,z13,z14) = (z14,z11,z12,z13)
and sums are straightforwardly defined using the '+' symbol as
summation:
   (1,0,0,0) + (0,1,0,0) = (1,1,0,0)
The system is sealed to the graphical representation via the excellent
upuesto
   (1,1,1,1) = (0,0,0,0).
This is the 3D mathematics that I propose is the generalization of the
complex plane in 3D; the four-signed numbers.

[jehovajah]

i have mentioned this before, but orientation is often overlooked due to over familiarity with the terms dimensions. This is why i use the notion of unary operator as opposed to just sign and generalise the notion of sign to signal. Using the coordinate system of notation also implies a vector space or preferably a tensor space rather than a number or rather numeral. These are fundamental confusions i am exploring and trying to get round.

Anyway your assertion of a 3d number system requires that the axes are non orthogonal. They have to lie in the four corners of a tetrahedron if they are to be regularly spaced, otherwise the four axes fit the xy plane and one can use sign on both pairs of opposing axes. The plane as it turns out is used by many paired variables to geometrise their locii. 

[Yannis]

Hello Kujonai and TimGolden,

I just discovered your previous post of 2009.
I proposed an algebra identical or very near of yours in 2011, we converge !
I thing that formalism is perhaps the best for a true (natural) generalization of complexes numbers a any dimensions, even if the pseudo-fractals calculated by that way are a little desappointing. It's certainly a new concept and representation of multicomplexes MCn.
See my post: "the ABSOLIEN numbers" on fractal forum : fractal maths/general discussion/

My proposition seems more simple because I use only vectors and matrix formalism of positive values unsigned (the sign simply corresponds to the position in the vector), and a very simple multiplication = descrete convolution. But it's sure that Tim Golden discribes the first that new formalism witch is equivalent to mine.
You can see my site on google too, in english or french:  https://sites.google.com/site/yannispicart/

Espero su opinion con mucho intéres.

[jehovajah]

Tim is about and infrequently active on this forum, but Kujonai i have not heard from for years. I suggest you message Tim if you want him to get back to you.
I have marked your thread as one to read as and when i get time Yannis, but that time is not just yet. Please do not feel ignored, Many of us are very busy and getting on with the outfall after the big Mandelbulb push when we spent so much of our lives collaborating that we nearly lost the balance in the rest of our lives!

This forum is truly of world class importance, the gateway into an understanding of our universe, and yet Chris struggles to keep it online and accessible! IT WAS EVER THUS, I AM AFRAID. THE ENTHUSIASM OF OUR MEMBERS has uncovered the deepest relationships we can know, but this is thought to be inconsequential because no one has figured out how to rob us of it by making big bucks from our findings.

I feel proud to be ablw to say "our" even though i did not originate or formulate the solutions, others did . Nevertheless the wealth of experise and understanding available in this forum is invaluable and i celebrate it and encourage all who can to donate what they can afford to kep it such a fun, creative artistic experimental place.